Abstract

Motivated by the separability problem in quantum systems 2⊗4, 3⊗3 and 2⊗2⊗2, we study the maximal (proper) faces of the convex body, S1, of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H=H1⊗H2⊗⋯⊗Hn. To any subspace V⊆H, we associate a face FV of S1 consisting of all states ρ∈S1 whose range is contained in V . We prove that FV is a maximal face if and only if V is a hyperplane. If V =|ψ〉⊥, where |ψ〉 is a product vector, we prove that DimFV=d2−1−∏(2di−1), where di=DimHi and d=∏di. We classify the maximal faces of S1 in the cases 2⊗2 and 2⊗3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2⊗2, and 20 and 24 for 2⊗3. The boundary, ∂S1, of S1 is the union of all maximal faces. When d>6, it is easy to show that there exist full states on ∂S1, i.e. states ρ∈∂S1 such that all partial transposes of ρ (including ρ itself) have rank d. Ha and Kye have recently constructed explicit such states in 2⊗4 and 3⊗3. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter b>0, b≠1. Each face in the family is a nine-dimensional simplex, and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses for these faces and analyse the three limiting cases b=0,1,∞.